Introduction
Consider a small insect moving on a thin sheet of paper that is gently floating in a calm stream. The paper itself is being carried by the flow of water at a steady rate of 10 cm per second in the forward direction. Meanwhile, the insect is moving crosswise on the paper, from a point \( A \) to a point \( B \), at a rate of 8 cm per second. The distance between \( A \) and \( B \) is exactly 8 cm, so the insect takes 1 second to travel from \( A \) to \( B \).
Now, during this 1 second, the sheet of paper does not remain stationary. It moves forward with the flow of water and covers a distance of 10 cm. The insect, from its own perspective, is simply moving in a straight path from \( A \) to \( B \), covering 8 cm. However, an observer standing on a bridge looking down at the stream sees something different. From this external point of view, the insect has not only moved 8 cm in the crosswise direction but has also been carried 10 cm forward by the moving paper.
To determine the actual distance the insect has traveled, we observe that its motion is composed of two movements occurring simultaneously:
- The movement of the insect on the paper, which is 8 cm in 1 second.
- The movement of the paper itself, which is 10 cm in 1 second.
Since these two movements occur perpendicularly to each other—one along the width of the paper and the other along the direction of the stream—the insect’s overall movement forms the diagonal of a right triangle where one side is 8 cm and the other is 10 cm. The total distance the insect has moved, as seen by an observer, is given by the Pythagorean theorem:
Thus, while the insect itself perceives that it has only moved 8 cm, an observer sees that it has actually traveled a longer diagonal path of approximately 12.81 cm in 1 second.
Now, suppose instead that the insect had been moving along the same direction as the paper’s motion, instead of crosswise. That is, if the insect were walking in the same forward direction as the moving paper at 8 cm per second, then in 1 second, it would have traveled 8 cm on the paper while the paper itself moved 10 cm forward. In this case, both motions occur in the same direction, so the total distance covered, as seen by the observer, is simply:
Thus, the way we observe the insect’s total movement depends not just on the magnitudes of the two motions but also on the direction in which they occur. When the two motions are perpendicular, they combine differently than when they are in the same direction.
This example illustrates an important idea: certain physical quantities depend not just on how much something moves but also on in which direction it moves. The movement of the insect and the movement of the paper are both meaningful, but their overall effect together depends on how these motions relate to each other in direction. We call such quantities vector quantities, meaning that both their magnitude (how much) and direction (which way) are essential in determining the final result.
Directed Line Segment
A vector quantity is represented geometrically as a directed line segment. A directed line segment is a segment of a straight line that has an initial point, a terminal point, and a specific direction. Unlike an ordinary line segment, which only has length, a directed line segment encodes additional information related to the concept of displacement, motion, or force.
Definition of a Directed Line Segment
A directed line segment is a line segment with an initial point and a terminal point, where the segment is oriented from the initial point toward the terminal point. If a point \( A \) is the initial point and a point \( B \) is the terminal point, we say that the directed line segment extends from \( A \) to \( B \), and we denote it as \( \overrightarrow{AB} \) or simply as \( \overrightarrow{v} \) is we give it a name \(v\).
Fundamental Characteristics of a Directed Line Segment
Every directed line segment possesses three fundamental characteristics:
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Length: The length of the directed line segment is the distance between its initial and terminal points. This length represents the magnitude of the vector quantity. If \( \overrightarrow{a} \) is a vector, its length is denoted by \( | \overrightarrow{a} | \), which gives the scalar measure of its magnitude.
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Line of Support: The infinite straight line on which the directed line segment lies is called its line of support. The vector quantity is completely constrained to this line in terms of direction.
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Sense: The sense of the directed line segment is determined by the order of its endpoints. If the segment extends from \( A \) to \( B \), its sense is from \( A \) to \( B \), and we write it as \( \overrightarrow{AB} \). If instead, it extended from \( B \) to \( A \), it would have the opposite sense, denoted as \( \overrightarrow{BA} \).
We can compare the sense of two vectors only if they have the same line of support or parallel lines of support. If two vectors do not lie on the same or parallel lines, the concept of comparing their sense becomes meaningless, as their directions exist in different spatial orientations. In such cases, attempting to determine whether they point "in the same direction" or "in opposite directions" is absurd because their lines of support do not allow for a meaningful comparison.
However, if two directed line segments share the same line of support or lie on parallel lines, their sense can be classified in two ways:
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Same Sense: If two directed line segments lie along the same or parallel lines and point in the same direction, they are said to have the same sense. That is, if \( \overrightarrow{AB} \) and \( \overrightarrow{CD} \) lie on the same or parallel lines and point in the same direction, then they have the same sense.
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Opposite Sense: If two directed line segments lie along the same or parallel lines but point in opposite directions, they have the opposite sense. This means that one vector is the negative of the other.
If the lines of support of two vectors are neither the same nor parallel, there is no meaningful way to compare their sense, and thus, any discussion of whether they are in the "same" or "opposite" direction is ill-posed.
In the above diagram, multiple directed line segments are shown, each with a well-defined initial point, terminal point, and direction. The relationships between these segments can be analyzed based on their lines of support and sense.
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\( \overrightarrow{AB} \) and \( \overrightarrow{CD} \) lie on parallel lines of support and point in the same direction. Since the sense of a directed line segment is determined by the order of its points, these two segments have the same sense.
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\( \overrightarrow{AB} \) and \( \overrightarrow{EF} \) also lie on parallel lines of support, but they point in opposite directions. This means that these two segments have opposite sense.
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\( \overrightarrow{AB} \) and \( \overrightarrow{GH} \) do not share the same line of support, nor do their lines of support run parallel. Since sense is only meaningful when comparing vectors along the same or parallel supports, the sense of \( \overrightarrow{AB} \) and \( \overrightarrow{GH} \) cannot be compared.
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\( \overrightarrow{GH} \) and \( \overrightarrow{IJ} \), however, do lie on the same line of support but point in opposite directions. Since their order is reversed relative to one another, they have opposite sense.
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Magnitude of a Vector
The magnitude (or length) of a vector is the distance between its initial point and terminal point in a directed line segment. If a vector is represented as a directed line segment \( \overrightarrow{AB} \), its magnitude is denoted as
and represents the length of the segment joining points \( A \) and \( B \). Since magnitude is a scalar quantity, it is always non-negative.
For any vector \( \overrightarrow{a} \), the magnitude satisfies the following properties:
- \( | \overrightarrow{a} | \geq 0 \), with equality holding if and only if \( \overrightarrow{a} \) is the null vector.
- A vector of unit length is called a unit vector, meaning \( | \hat{a} | = 1 \).
Null Vector
A null vector (or zero vector) is a vector whose magnitude is zero. It is denoted by \( \overrightarrow{0} \) and satisfies:
A null vector has no definite direction since its initial and terminal points coincide. Thus, \(\overrightarrow{AA}\) is a null vector. Geometrically, it is represented as a directed line segment of zero length, meaning it does not contribute to displacement.
Direction of a Vector
The direction of a vector is determined by two essential properties:
- Line of Support – the infinite line along which the vector lies.
- Sense – the orientation from the initial point to the terminal point.
A vector's direction is fully defined only when both of these characteristics are specified.
Comparing Directions of Two Vectors
Two vectors have the same direction if and only if:
- Their lines of support are either the same or parallel.
- Their sense is the same.
If either of these conditions is not satisfied, the vectors have different directions. In particular:
- If the vectors have parallel lines of support but opposite sense, they are said to be oppositely directed.
- If the vectors do not even have parallel lines of support, their directions are entirely different.
Thus, the direction of a vector is fully determined by its line of support and sense, and any violation of these conditions results in distinct directions.
In the diagram, the directed line segments exhibit relationships based on their lines of support and sense. The segments \( \overrightarrow{AB} \) and \( \overrightarrow{GH} \) have the same direction since their lines of support are parallel and their sense is the same. Similarly, \( \overrightarrow{CD} \) and \( \overrightarrow{EF} \) have the same direction due to their parallel support and matching sense. However, \( \overrightarrow{AB} \) and \( \overrightarrow{CD} \) are oppositely directed, meaning their directions are different. The segments \( \overrightarrow{AB} \) and \( \overrightarrow{KL} \) do not share the same or parallel lines of support, so their directions are distinct. Finally, \( \overrightarrow{IJ} \) and \( \overrightarrow{KL} \) have the same direction as they lie on parallel lines with the same sense.
Representing a Vector Quantity as a Direct Line Segment
To represent a vector quantity as a directed line segment, we must translate its magnitude and direction into a geometric form that preserves both characteristics. A vector quantity such as velocity, force, or displacement has both size (magnitude) and orientation (direction). On paper, we represent such quantities as directed line segments where:
- The length of the segment is proportional to the magnitude of the vector: A suitable scale is chosen to map physical magnitudes to measurable distances on paper.
- The segment is drawn along the specified direction: The segment must be placed so that it aligns with the given direction in space.
- The sense of the segment is from the initial point toward the terminal point: This ensures the correct orientation of the vector.
Example: Representing Velocity as a Directed Line Segment
Consider two cars moving in different directions:
- Car A is moving in a straight line at a speed of 40 km/h in the northeast (NE) direction.
- Car B is moving in a straight line at a speed of 50 km/h in the south (S) direction.
The quantity speed alone is a scalar—it tells us how fast an object moves but gives no information about its direction. However, when speed is combined with direction, it forms velocity, which is a vector quantity.
To represent these velocities as directed line segments on paper, we choose a scale. Suppose we decide that:
Then, the velocity of Car A, being 40 km/h, is represented by a directed line segment of length:
pointing towards the northeast (NE) direction.
Similarly, the velocity of Car B, being 50 km/h, is represented by a directed line segment of length:
pointing towards the south (S) direction.
These directed line segments completely describe the vector quantities corresponding to the velocities of the cars. They capture both the magnitude (through their length) and the direction (through their orientation).
The Next Step
Now that we have translated these velocity vectors into directed line segments, we can begin analyzing their effects. What we do with these directed segments—whether we compare them, combine them, or decompose them—will be the subject of our discussion in the next section.